Question: Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, and $C$ is never immediately followed by $A$. How many seven-letter good words are there?
Solution: There are three choices for the first letter and two choices for each subsequent letter, so there are $3\cdot2^{n-1}\ n$-letter good words.  Substitute $n=7$ to find there are $3\cdot2^6=\boxed{192}$ seven-letter good words.